Problem: What is the sum of all integer values $n$ for which $\binom{20}{n}+\binom{20}{10}=\binom{21}{11}$?
Answer: From Pascal's identity $\binom{n-1}{k-1}+\binom{n-1}{k}=\binom{n}{k}$.

Therefore, we have $\binom{20}{11}+\binom{20}{10}=\binom{21}{11}$, so $n=11$.

We know that $\binom{21}{11}=\binom{21}{21-11}=\binom{21}{10}$.

We use Pascal's identity again to get $\binom{20}{9}+\binom{20}{10}=\binom{21}{10}$, so $n=9$.

There are two values for $n$, $9$ and $11$, so the sum is $9+11=\boxed{20}$.